CHAPTER 5 SUBALGEBRAS AND UNIONS §a Weak and relative subalgebras The subalgebra concept has two very natural weakenings. The first of these concerns the algebraic structure that is inherited by arbitrary subsets of the universe of an algebra, as opposed to subuniverses; one of the more compel- ling ways in which partial algebras arise in the study of total algebras. The second corresponds with the algebraic structure necessary for a mapping from another algebra to be a homomorphism. a1 Definition. Let A = 〈A, I〉 and B = 〈B, J〉 be algebras such that A ⊆ B. (a) A is a relative subalgebra of B, notation A ⊆ B, and B a outer extension of A, if Dom(I) = Dom(J), and for every symbol S ∈ Dom(J), I(S) = J(S)A. (b) A is a weak subalgebra of B, notation A ⊆w B, if Dom(I) = Dom(J), and for every symbol S ∈ Dom(I), I(S) ⊆ J(S). Observe that any subset A of the universe of an algebra B determines a unique relative subalgebra of B that has A for universe. We denote it by AB. On the other hand, there may be many weak subalgebras of B that have A for a universe. In particular, there may be many weak subalgebras of B that have B for a universe; we shall call them weakenings of B. If A is a weakening of B, then B is an inner extension of A. If B is a total algebra, and A is dense in B, we say that B is a completion of AB. Examples A (a) Suppose A is a weak subalgebra of B. Then 1B is an embedding of A into B. A (b) Conversely, if 1B is a homomorphism of A into B, and NomA = NomB, then A is a weak subalgebra of B. (c) Let K be a field. Consider the weakening V of Kn in which a · b is defined only if a ∈ {k}n for some k ∈ K. Then V is, essentially, an n-dimensional vector space over K. (d) I als relatieve subalgebra van een algebra met oneindige intervallen. (e) Concrete categories (see 4…) seldom are weak subalgebras of Set. For ex- ample, different algebras may have the same universe; and a fortiori different homomorphisms may reduce to the same function. a2 Proposition. (i) Let A and B be algebras. If A ≤ B, then also A ⊆ B; and if A ⊆ B, then A ⊆w B. (ii) If A ⊆ B, then A ≤ B if and only if A is a subuniverse of B. In particular, if all the operations of B are predicates, A ⊆ B if and only if A ≤ B. (iii) The [weak/relative] subalgebras of any given algebra form a set. 5. SUBALGEBRAS Proof of (iii). By (i) and Comprehension, we need only show that the weak subalgebras form a set. But these may be considered a subset of PA × ∏ P(J(S)) . S∈Dom(J) 25.3 Theorem. Let A be a weak subalgebra of B, and f: B —–› C a homomor- phism. Then f |`A is a homomorphism from A to C. A Proof. Observe that f |`A = f 1B; then use 23.2(ii) and example (b) under De- finition 1. A4 Theorem. For each i ∈ I, let Bi be a relative subalgebra of Ai. Then ∏Bi ⊆ ∏Ai. i ∈I i ∈I Proof. Put A := ∏iAi and B := ∏iBi. Suppose b0,…, bn ∈ B. Then A Ai Q (b1,…, bn) = b0 ⇔ ∀i ∈ I Q (b1(i),…, bn(i)) = b0(i) Bi ⇔ ∀i ∈ I Q (b1(i),…, bn(i)) = b0(i), since Bi ⊆ Ai and b0(i),…, bn(i) ∈ Bi B ⇔ Q (b1,…, bn) = b0. §26. Union and Intersection If A = 〈A, I〉 and B = 〈B, J〉 are relational structures, we can define their union A ∪ B as 〈A ∪ B, K〉, where Dom(K) = Dom(I) ∪ Dom(J) and for every K ∈ Dom(K), K(R) = RA ∪ RB. The generalization to arbitrary families of relational structures is straightforward. This approach will not work for structures with operations, as long as we hold on to the restriction that opera- tions be single-valued. We shall have to presuppose some kind of hierarchy in the structures to be combined, so that we are able to decide which has prece- dence in case of conflict. In principle, the order of precedence could be differ- ent for different conflicts, but we shall have no use for this ultimate subtlety. 26.1 Definition. Let A = 〈Aξ|ξ < α〉 be a sequence of algebras, Aξ = 〈Aξ, Iξ〉. The union of A is the algebra B = 〈B, J〉 where B = Aξ, Dom(J) = Dom(Iξ), ξ<α ξ<α and for every Q ∈ Dom(J), Dom(J(Q)) = Dom(Iξ(Q)), ξ<α and if ξ is the first index such that b ∈ Dom(Iξ(Q)), J(Q)(b) = Iξ(Q)(b). We denote the union of a sequence 〈Aξ| ξ < α〉 of algebras by Aξ. ξ<α Algebras A and B are compatible if for every operation symbol Q, for every sequence d ∈ (A ∩ B)*, if both QA(d)↓ and QB(d)↓, then QA(d) = QB(d). 72 14 januari 2014 An enumeration of a set X is a mapping of an ordinal onto X. In other words, it is a listing of all the elements of X in a sequence, with repetitions al- lowed. 26.2 Lemma. Let 〈Ai|i ∈ I〉 be a family of pairwise compatible algebras; sup- pose 〈Aξ|ξ < α〉 and 〈Bη|η < β〉 are enumerations of {Ai|i ∈ I}. Then Aξ = Bη. ξ<α η<β Proof. Let C := Aξ and D := Bη. ξ<α η<β The universes C and D are obviously the same. Let Q be an operation symbol, and suppose c is a sequence of elements of C such that QC(c)↓. By definition, QC(c) = QAγ(c), where γ is the least ξ such that QAξ(c)↓. Now if δ is the least B η such that Q η(c)↓ — such η certainly exist, since Aγ occurs in the enumera- B A tion 〈Bη|η < β〉 — then since Aγ and Bδ are compatible, Q δ (c) = Q γ(c). Hence QD(c) = QC(c). By this lemma, we may define the union iAi , for a family 〈Ai|i ∈ I〉 of pairwise compatible algebras, as (Aξ|ξ < α) for any enumeration 〈Aξ〉ξ<α of {Ai|i ∈ I}. 26.3 Corollary. Let 〈Aj|j ∈ J〉 be a family of weak subalgebras of an algebra B. Then Aj is a weak subalgebra of B. j∈J Proof. The union exists: since the operations of the Aj are contained in opera- tions of B, the Aj are pairwise compatible. 26.4 Proposition. If 〈Ai|i ∈ I〉 is a family of pairwise compatible structures, then the canonical injection a —| › a of any Ai into iAi is an embedding of Ai into iAi. 26.5 Definition. The intersection iAi of a family 〈Ai|i ∈ I〉 of algebras is the algebra B = 〈B, K〉 where B = iAi, and K is the function on iDomIAi de- fined, for every symbol S in its domain, by K(S) = iIAi(S). The canonical injection of iAi into any Ai is an embedding of iAi into Ai. Exercises §25 1. Verify that the relative subalgebra relation and the weak subalgebra relation are orderings of the class of all algebras. 2. Verify Example (b). 2. Prove (i) and (ii) of Proposition 2. §26 1. Is the union of two orders necessarily an order? 2. Prove Proposition 4. 73 5. SUBALGEBRAS A unary algebra is an algebra with only unary basic operations. The algebra 〈, S〉 of the natural numbers with the successor operation is a prime example. 3. Let B be a unary algebra, and 〈Aj|j ∈ J〉 a family of subalgebras of B. Prove that jAj ≤ B. 4. (a) Show by example that in general the union of a family of subalgebras of an algebra B need not even be a relative subalgebra of B. (b) Show by example that the union of a family of relative subalgebras of a unary algebra B need not be a relative subalgebra of B. 5. Let A = {Aj|j ∈ J} be a collection of subsets of the universe of an algebra B. Assume that A is directed by inclusion, that is, 〈A, ⊆〉 is a directed order. Put Aj := (Aj)B. (a) Prove that jAj ⊆ B. (b) Prove that jAj ≤ B if all the Aj are subalgebras of B. 74 .